semPower.genSigma: semPower.genSigma

Description

Generate a covariance matrix (and a mean vector) and associated lavaan model strings based on CFA or SEM model matrices.

Usage

semPower.genSigma(
  Lambda = NULL,
  Phi = NULL,
  Beta = NULL,
  Psi = NULL,
  Theta = NULL,
  tau = NULL,
  Alpha = NULL,
  ...
)

Value

Returns a list (or list of lists for multiple group models) containing the following components:

Sigma: implied variance-covariance matrix.
mu: implied means
Lambda: loading matrix
Phi: covariance matrix of latent variables
Beta: matrix of regression slopes
Psi: residual covariance matrix of latent variables
Theta: residual covariance matrix of observed variables
tau: intercepts
Alpha: factor means
modelPop: lavaan model string defining the population model
modelTrue: lavaan model string defining the "true" analysis model freely estimating all non-zero parameters.
modelTrueCFA: lavaan model string defining a model similar to modelTrue, but purely CFA based and thus omitting any regression relationships.

Arguments

Lambda: factor loading matrix. A list for multiple group models. Can also be specified using various shortcuts, see genLambda().
Phi: for CFA models, factor correlation (or covariance) matrix or single number giving the correlation between all factors or NULL for uncorrelated factors. A list for multiple group models.
Beta: for SEM models, matrix of regression slopes between latent variables (all-y notation). A list for multiple group models.
Psi: for SEM models, variance-covariance matrix of latent residuals when Beta is specified. If NULL, a diagonal matrix is assumed. A list for multiple group models.
Theta: variance-covariance matrix between manifest residuals. If NULL and Lambda is not a square matrix, Theta is diagonal so that the manifest variances are 1. If NULL and Lambda is square, Theta is 0. A list for multiple group models.
tau: vector of intercepts. If NULL and Alpha is set, these are assumed to be zero. If both Alpha and tau are NULL, no means are returned. A list for multiple group models.
Alpha: vector of factor means. If NULL and tau is set, these are assumed to be zero. If both Alpha and tau are NULL, no means are returned. A list for multiple group models.
...: other

Details

This function generates the variance-covariance matrix of the $p$ observed variables $\Sigma$ and their means $\mu$ via a confirmatory factor (CFA) model or a more general structural equation model.

In the CFA model, $$\Sigma = \Lambda \Phi \Lambda' + \Theta$$ where $\Lambda$ is the $p \cdot m$ loading matrix, $\Phi$ is the variance-covariance matrix of the $m$ factors, and $\Theta$ is the residual variance-covariance matrix of the observed variables. The means are $$\mu = \tau + \Lambda \alpha$$ with the $p$ indicator intercepts $\tau$ and the $m$ factor means $\alpha$.

In the structural equation model, $$\Sigma = \Lambda (I - B)^{-1} \Psi [(I - B)^{-1}]' \Lambda' + \Theta $$ where $B$ is the $m \cdot m$ matrix containing the regression slopes and $\Psi$ is the (residual) variance-covariance matrix of the $m$ factors. The means are $$\mu = \tau + \Lambda (I - B)^{-1} \alpha$$

In either model, the meanstructure can be omitted, leading to factors with zero means and zero intercepts.

When $\Lambda = I$, the models above do not contain any factors and reduce to ordinary regression or path models.

If Phi is defined, a CFA model is used, if Beta is defined, a structural equation model. When both Phi and Beta are NULL, a CFA model is used with $\Phi = I$, i. e., uncorrelated factors. When Phi is a single number, all factor correlations are equal to this number.

When Beta is defined and Psi is NULL, $\Psi = I$.

When Theta is NULL, $\Theta$ is a diagonal matrix with all elements such that the variances of the observed variables are 1. When there is only a single observed indicator for a factor, the corresponding element in $\Theta$ is set to zero.

Instead of providing the loading matrix $\Lambda$ via Lambda, there are several shortcuts (see genLambda()):

loadings: defines the primary loadings for each factor in a list structure, e. g. loadings = list(c(.5, .4, .6), c(.8, .6, .6, .4)) defines a two factor model with three indicators loading on the first factor by .5, , 4., and .6, and four indicators loading in the second factor by .8, .6, .6, and .4.
nIndicator: used in conjunction with loadM or loadMinmax, defines the number of indicators by factor, e. g., nIndicator = c(3, 4) defines a two factor model with three and four indicators for the first and second factor, respectively. nIndicator can also be a single number to define the same number of indicators for each factor.
loadM: defines the mean loading either for all indicators (if a single number is provided) or separately for each factor (if a vector is provided), e. g. loadM = c(.5, .6) defines the mean loadings of the first factor to equal .5 and those of the second factor do equal .6
loadSD: used in conjunction with loadM, defines the standard deviations of the loadings. If omitted or NULL, the standard deviations are zero. Otherwise, the loadings are sampled from a normal distribution with N(loadM, loadSD) for each factor.
loadMinMax: defines the minimum and maximum loading either for all factors or separately for each factor (as a list). The loadings are then sampled from a uniform distribution. For example, loadMinMax = list(c(.4, .6), c(.4, .8)) defines the loadings for the first factor lying between .4 and .6, and those for the second factor between .4 and .8.

Examples

Run this code

if (FALSE) {
# single factor model with five indicators each loading by .5
gen <- semPower.genSigma(nIndicator = 5, loadM = .5)
gen$Sigma     # implied covariance matrix
gen$modelTrue # analysis model string
gen$modelPop  # population model string

# orthogonal two factor model with four and five indicators each loading by .5
gen <- semPower.genSigma(nIndicator = c(4, 5), loadM = .5)

# correlated (r = .25) two factor model with 
# four indicators loading by .7 on the first factor 
# and five indicators loading by .6 on the second factor
gen <- semPower.genSigma(Phi = .25, nIndicator = c(4, 5), loadM = c(.7, .6))

# correlated three factor model with variying indicators and loadings, 
# factor correlations according to Phi
Phi <- matrix(c(
  c(1.0, 0.2, 0.5),
  c(0.2, 1.0, 0.3),
  c(0.5, 0.3, 1.0)
), byrow = TRUE, ncol = 3)
gen <- semPower.genSigma(Phi = Phi, nIndicator = c(3, 4, 5), 
                         loadM = c(.7, .6, .5))

# same as above, but using a factor loadings matrix
Lambda <- matrix(c(
  c(0.8, 0.0, 0.0),
  c(0.7, 0.0, 0.0),
  c(0.6, 0.0, 0.0),
  c(0.0, 0.7, 0.0),
  c(0.0, 0.8, 0.0),
  c(0.0, 0.5, 0.0),
  c(0.0, 0.4, 0.0),
  c(0.0, 0.0, 0.5),
  c(0.0, 0.0, 0.4),
  c(0.0, 0.0, 0.6),
  c(0.0, 0.0, 0.4),
  c(0.0, 0.0, 0.5)
), byrow = TRUE, ncol = 3)
gen <- semPower.genSigma(Phi = Phi, Lambda = Lambda)

# same as above, but using a reduced loading matrix, i. e.
# only define the primary loadings for each factor
loadings <- list(
  c(0.8, 0.7, 0.6),
  c(0.7, 0.8, 0.5, 0.4),
  c(0.5, 0.4, 0.6, 0.4, 0.5)
)
gen <- semPower.genSigma(Phi = Phi, loadings = loadings)

# Provide Beta for a three factor model
# with 3, 4, and 5 indicators 
# loading by .6, 5, and .4, respectively.
Beta <- matrix(c(
                c(0.0, 0.0, 0.0),
                c(0.3, 0.0, 0.0),  # f2 = .3*f1
                c(0.2, 0.4, 0.0)   # f3 = .2*f1 + .4*f2
               ), byrow = TRUE, ncol = 3)
gen <- semPower.genSigma(Beta = Beta, nIndicator = c(3, 4, 5), 
                         loadM = c(.6, .5, .4))

# two group example: 
# correlated two factor model (r = .25 and .35 in the first and second group, 
# respectively)
# the first factor is indicated by four indicators loading by .7 in the first 
# and .5 in the second group,
# the second factor is indicated by five indicators loading by .6 in the first 
# and .8 in the second group,
# all item intercepts are zero in both groups, 
# the latent means are zero in the first group
# and .25 and .10 in the second group.
gen <- semPower.genSigma(Phi = list(.25, .35), 
                         nIndicator = list(c(4, 5), c(4, 5)), 
                         loadM = list(c(.7, .6), c(.5, .8)), 
                         tau = list(rep(0, 9), rep(0, 9)), 
                         Alpha = list(c(0, 0), c(.25, .10))
                         )
gen[[1]]$Sigma  # implied covariance matrix group 1 
gen[[2]]$Sigma  # implied covariance matrix group 2
gen[[1]]$mu     # implied means group 1 
gen[[2]]$mu     # implied means group 2
}

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